Derivations and identities for Fibonacci and Lucas polynomials

TL;DR

This paper introduces Fibonacci and Lucas derivations in polynomial algebras, establishing that their kernels define polynomial identities for Fibonacci and Lucas polynomials, and linking identities for Appel polynomials to these sequences.

Contribution

It presents a novel framework of derivations for Fibonacci and Lucas polynomials and connects identities across different polynomial families.

Findings

Kernel of derivations defines polynomial identities

Identities for Appel polynomials translate to Fibonacci and Lucas

Intertwining maps relate different polynomial identities

Abstract

We introduce the notion of Fibonacci and Lucas derivations of the polynomial algebras and prove that any element of kernel of the derivations defines a polynomial identity for the Fibonacci and Lucas polynomials. Also, we prove that any polynomial identity for Appel polynomial yields a polynomial identity for the Fibonacci and Lucas polynomials and describe the corresponding intertwining maps.